question_answer

The area of the region bounded by the curves \[y=ex,\text{ y}=lo{{g}_{e}}x\]and lines\[x=1,\text{ }x=2\]is

A)  \[{{(e-1)}^{2}}\]

B)  \[{{e}^{2}}-e+1\]

C)  \[{{e}^{2}}-e+1-2lo{{g}_{e}}2\]

D)  \[{{e}^{2}}+e-2lo{{g}_{e}}2\]

Correct Answer: C

Solution :

 Required area, \[A=\int_{1}^{2}{({{e}^{x}}-{{\log }_{e}}x)}\,dx\] \[=[{{e}^{x}}]_{1}^{2}-[x{{\log }_{e}}x-\int{1\,dx}]_{1}^{2}\] \[={{e}^{2}}-e-[x{{\log }_{e}}x-x]_{1}^{2}\] \[={{e}^{2}}-e-[2{{\log }_{e}}2-2(0-1)]\] \[={{e}^{2}}-e-2{{\log }_{e}}2+1\]